nLab bubble of nothing





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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


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A bubble of nothing in the sense of Witten 82 is an instability of the Kaluza-Klein vacuum where a gravitational instanton mediates the decay of a 55-dimensional Euclidean Schwarzschild spacetime into nothing: the hole in space grows at the speed of light until it hits null infinity?, leading to the “annihilation” of spacetime.

Construction of the solution

Let us start from the spacetime ( 4B R 3)×S 1(\mathbb{R}^{4}-B_R^3)\times S^1, where B R 3B_R^3 is a ball of radius RR and S 1S^1 is a circle of radius R 5R_5, equipped with the following Euclidean Schwarzschild metric:

g=r 2vol S 3+dr 21R 2r 2+R 5 2(1R 2r 2)dθ 2, g \;=\; r^2\mathrm{vol}_{S^3} + \frac{\mathrm{d}r^2}{1-\frac{R^2}{r^2}} + R_5^2 \left(1-\frac{R^2}{r^2}\right)\mathrm{d}\theta^2,

where θ\theta is the coordinate on the circle and rr is the radius of 4\mathbb{R}^{4}.

Let the volume of the 33-sphere be vol S 3=dχ 2+sin(χ)vol S 2\mathrm{vol}_{S^3} = \mathrm{d}\chi^2 + \sin(\chi)\mathrm{vol}_{S^2}. We can now go back to the Minkowski signature by the Wick rotation ψ:=iχ+iπ2\psi := -i\chi + i\frac{\pi}{2}. Thus, we have

g=r 2dψ 2+dr 21R 2r 2+r 2cosh 2(ψ)vol S 2+R 5 2(1R 2r 2)dθ 2. g \;=\; -r^2\mathrm{d}\psi^2 + \frac{\mathrm{d}r^2}{1-\frac{R^2}{r^2}} + r^2\cosh^2(\psi)\mathrm{vol}_{S^2} + R_5^2 \left(1-\frac{R^2}{r^2}\right)\mathrm{d}\theta^2.


By change of coordinates ρ:=rcosh(ψ)\rho := r\cosh(\psi) and t:=rsinh(ψ)t:=r\sinh(\psi), we see that at large radius the metric goes to the Minkowski metric

lim rg=dt 2+dρ 2+ρ 2vol S 2+R 5 2dθ 2.\lim_{r\rightarrow \infty}g \;=\; -\mathrm{d}t^2 +\mathrm{d}\rho^2 + \rho^2\mathrm{vol}_{S^2}+R_5^2\mathrm{d}\theta^2.

However, the coordinates (t,ρ)(t,\rho) parametrize all the 44-dimensional base manifold but the region {ρ 2t 2R}\{ \rho^2 - t^2 \leq R \} of the bubble. The boundary of the bubble in the 44-dimensional base space has a radius which grows with time as

ρ BON(t)=R 2+t 2. \rho_{\mathrm{BON}}(t) \;=\;\sqrt{R^2+t^2}.

Moreover, the effective size of the circle compactification will be depend on rr by

R 5(r)=R 51R 2r 2, R_{5}(r) \;=\; R_5\sqrt{1-\frac{R^2}{r^2}},

so that it goes to 00 for rRr\rightarrow R and it goes to R 5R_5 for for rr\rightarrow \infty.


Last revised on April 21, 2021 at 11:48:41. See the history of this page for a list of all contributions to it.